Analytical Solution of the Nonlinear Relativistic Boltzmann Equation

TL;DR

This paper derives an exact analytical solution to the nonlinear relativistic Boltzmann equation for a massless gas, revealing insights into scattering angle dependence and equilibrium stability.

Contribution

It introduces a novel analytical approach for solving the nonlinear relativistic Boltzmann equation with anisotropic scattering, connecting it to classical nonrelativistic solutions.

Findings

Analytical solution for a massless relativistic gas obtained.

Demonstrates stable equilibrium fixed point.

Clarifies the non-existence of BKW solutions for massive particles.

Abstract

We provide an exact analytical solution to the nonlinear relativistic Boltzmann equation for a homogeneous, anisotropically scattering massless gas. Utilizing a BKW-like trial solution, we cast the Boltzmann equation into a set of nonlinear coupled equations for scalar moments, based on which the analytical solution is derived. One remarkable feature of our analytical solution lies in the nontrivial scattering angle dependence. We also show that this analytical solution admits a stable fixed point corresponding to the equilibrium solution as long as the parameters are physically feasible. Furthermore, a clear correspondence between our solution and the BKW solution pertaining to nonrelativistic Maxwell molecules is established, thereby clarifying the non-existence of a BKW-type solution in the relativistic domain for massive particles.